Let $G$ be a topological group. A representation of $G$ in a Banach space $V$ is a continuous function $G\times V\to V$.
This MSE question asks about a possible categorical framework for representations of topological groups. As remarked the notion there recovers norm-continuous representations, which are emphasized here as distinct from the definition above.
Is there a pleasant categorical framework for representations in the above sense? In the discrete case, the perspectives of functors from $\mathbf BG$ and of modules over group algebras are very satisfying.
Added. I tried to read this paper by Bernstein, but I understood very little. As a novice to representation theory I am very interested in having "correct" definitions, from a maximally geometric perspective. Any references would be appreciated. I don't mind highly categorical language as long as it's rigorous and recovers the notion of continuous representation described above.
(It would be nice to have a formal version of the definition Bernstein proposes.)
